Self-intersections in combinatorial topology: statistical structure (Q421024)

The authors provide statistical information on the number of self-intersections of randomly chosen oriented closed curves on a given closed orientable surface. Each such curve can be represented as a word with respect to a fixed set of generators of the fundamental group. The main result says that the distribution of the number of self-intersections of a closed oriented curve of fixed word length (uniformly chosen at random) is asymptotically Gaussian. The quite accessible proof employs U-statistics of a suitably crafted Markov chain.